The actual proof of these takes quite a bit of math (maybe i just have not found the quick way to do it) and I’m not posting that here, but you can intuitively understand the formulas too.

The amount of active risk you want to take should be positively correlated with your information ratio. This is easy to understand, because the information ratio just measures your active return (which you want more of) per unit of active risk. So when IR increases you should be willing to take on more active risk, because you are better compensated for it. Also your active risk should be negatively correlated to the sharpe ratio of your benchmark (1/SR_b), because the higher the sharpe ratio of the benchmark the better of you will be just investing in the benchmark instead of taking additional active risk. Finally your optimal amount of active risk should increase (decrease) as the standard deviation of benchmark return increases (decreases). If your benchmark has a low standard deviation of returns you are once again comparatively better of investing in the benchmark than taking active risk and vice versa for a high standard deviation of benchmark returns.

The information ration measures how much value you add by taking on active risk. As such the higher IR the higher should your sharpe ratio be. (Unfortunately I do not have an easy explanation to remember the squares)

Memorize baby! You ain’t gon get time to derive anything in the exam. It’s like knowing P/B = (ROE-g)/(k-g). You can derive it, but ain’t nobody got time for that.